Compact attractors of an antithetic integral feedback system have a simple structure

Abstract

Since its introduction by Briat, Gupta and Khammash, the antithetic feedback controller design has attracted considerable attention in both theoretical and experimental systems biology. The case in which the plant is a twodimensional linear system (making the closed-loop system a nonlinear four-dimensional system) has been analyzed in much detail. This system has a unique equilibrium but, depending on parameters, it may exhibit periodic orbits. An interesting open question is whether other dynamical behaviors, such as chaotic attractors, might be possible for some parameter choices. This note shows that, for any parameter choices, every bounded trajectory satisfies a Poincaré-Bendixson property. The analysis is based on the recently introduced notion of k-cooperative dynamical systems. It is shown that the model is a strongly 2-cooperative system, implying that the dynamics in the omegalimit set of any precompact solution is conjugate to the dynamics in a compact invariant subset of a two-dimensional Lipschitz dynamical system, thus precluding chaotic and other strange attractors. Index TermsPoincaré-Bendixson property, k-cooperative dynamical systems, sign-regular matrices, synthetic biology, antithetic feedback, synthetic biology, nonlinear control theory.MM is with the School of EE-Systems, Tel Aviv University, Israel 69978.

Guidelines for designing the antithetic feedback motif

“…In [38,39], the authors determined analytic conditions on the reaction rate parameters of the antithetic feedback network such that the linearized closed loop system is stable. Subsequent research also discussed the stability of the nonlinear dynamics of closed loop antithetic feedback network [44,45]. Without stability, the antithetic feedback control would not be able to track the reference signal; instead, it would oscillate indefinitely.…”

“…Indeed, in the previous synthetic biological implementations of antithetic feedback in E. coli and S. cerevisiae, the antithetic feedback controller's relative steady-state error (as normalized to the reference) has been measured to be 5-50% [28,31]. Another consideration is the stability of the closed-loop antithetic feedback system since theoretical studies have demonstrated that, depending on the reaction rate parameters, the antithetic controller can become unstable and periodic oscillations can arise [39,42,44,45]. Therefore, it is important to consider how the stability, robustness, and steady-state error (performance) of the antithetic feedback motif depend on its synthetic biological implementation and to study in depth how to tune these properties [28,46].…”

Restoring circadian gene profiles in clock networks using synthetic feedback control

“…As a final remark, even though the design framework we describe here is for circadian clocks, the approach presented is potentially applicable to tracking or restoring any biological system characterized by entrainable, periodic oscillations, for which theoretical developments are garnering great interest (see e.g., 58,59 ).…”